Coarse embeddings of quotients by finite group actions

We prove that for a metric space X X and a finite group G G acting on X X by isometries, if X X coarsely embeds into a Hilbert space, then so does the quotient X / G X/G. A crucial step towards our main result is to show that for any integer k > 0 k > 0 the space of unordered k k -tuples of points in Hilbert space, with the 1 1 -Wasserstein distance, itself coarsely embeds into Hilbert space. Our proof relies on establishing bounds on the sliced Wasserstein distance between empirical measures in R n R^n.